For each of the following statements, indicate whether it is True or False: If U
ID: 3406281 • Letter: F
Question
For each of the following statements, indicate whether it is True or False: If U is an echelon form of A then the eigenvalues of U will be the same as the eigenvalues of A. If a matrix B is obtained from a matrix A using elementary row replacement operations only (no scaling, no row swapping, only adding a multiple of another row to a given row), then A and B will have the same eigenvalues. If two matrices A and B have the same determinant, then they will have the same eigenvalues. If lambda_1, and lambda_2, are distinct eigenvalues of a matrix A, v_1 is any vector in the eigenspace of A corresponding to lambda_1, and v_2 is any vector in the eigenspace of A corresponding to lambda_2, then v_1 and v_2 will be linearly independent. If v_1 and v_2 are linearly independent eigenvectors of a matrix A, then v_1 and v_2 are in different eigenspaces corresponding to distinct eigenvalues of A. If an n times n matrix A is diagonalizable, then every vector in R^n will be an eigenvector of A. If every vector in R^n can be expressed as a linear combination of the eigenvectors of A. then A is diagonalizable. If an n times n matrix A is diagonalizable, then, taken together, the basis vectors of each of the eigenspaces of A will span R^n. If two matrices A and B are similar, then they will have the same eigenvalues. If two matrices A and B have the same eigenvalues, then they will be similar. If two matrices A and B are similar, then they will have the same eigenvectors.Explanation / Answer
a) False
Row reducing changes the eigenvectors and eigenvalues.
b) False
Row reducing changes the eigenvectors and eigenvalues.
c) False
If 2 matrices A and B has same determinant, then the products of their eigenvalues (counted with the multiplicity in the algebraic closure of the field over which matrices are defined) are the same. So, for 2 pairs of different numbers, the product may be same. For eq: matrix with eigen values 4 and 3 has a product of 12 and matrix with eigenvalues 2 and 6 also has a product of 12.
d) True
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